Abstract
Algorithms for nonlinear programming and variational inequality problems are, in general, only guaranteed to converge in the limit to a Karush-Kuhn-Tucker point, in the case of nonlinear programs, or to a solution in the case of variational inequalities. In this paper, we derive sufficient conditions for nonlinear programs with convex feasible sets such that any convergent algorithm can be modified, by adding a convex subproblem with a linear objective function, to guarantee finite convergence in a generalized sense. When the feasible set is polyhedral, the subproblem is a linear program and finite convergence is obtained. Similar results are also developed for variational inequalities.
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Communicated by O. L. Mangasarian
The research of the first author was supported in part by the Office of Naval Research under Contract No. N00014-86-K-0173.
The authors are indebted to Professors Olvi Mangasarian, Garth McCormick, Jong-Shi Pang, Hanif Sherali, and Hoang Tuy for helpful comments and suggestions and to two anonymous referees for constructive remarks and for bringing to their attention the results in Refs. 13 and 14.
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Al-Khayyal, F., Kyparisis, J. Finite convergence of algorithms for nonlinear programs and variational inequalities. J Optim Theory Appl 70, 319–332 (1991). https://doi.org/10.1007/BF00940629
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DOI: https://doi.org/10.1007/BF00940629